### Abstract

We focus on an extension of the assignment problem with additional conflict (pair) constraints in conjunction with the assignment constraints and binary restrictions. Given a bipartite graph with a cost associated with each edge and a conflict set of edge pairs, the assignment problem with conflict constraints corresponds to finding a minimum weight perfect matching without any conflicting edge pair. For example, some chemicals cannot be processed on close processors, food and toxic products cannot be stored neighboring locations at the same storage area, and machines cannot be sent to process jobs without satisfying some spatial constraints. Unlike the well-known assignment problem, this problem is NP-hard. We first introduce a realistic special class and demonstrate its polynomial solvability. Then, we propose a Branch-and-Bound algorithm and a Russian Doll Search algorithm using the assignment problem relaxations for lower bound computations, and introduce combinatorial branching rules based on the conflicting edges in an optimal solution of the relaxations. According to the extensive computational experiments we can say that the proposed algorithms are very efficient.

Original language | English |
---|---|

Pages (from-to) | 214-229 |

Number of pages | 16 |

Journal | Computers & Operations Research |

Volume | 111 |

DOIs | |

Publication status | Published - Nov 2019 |

### Fingerprint

### Keywords

- Assignment Problem
- Integer programming
- Branch and bound
- Conflicts
- Branch-and-bound
- Assignment problem

### Cite this

*Computers & Operations Research*,

*111*, 214-229. https://doi.org/10.1016/j.cor.2019.07.001

}

*Computers & Operations Research*, vol. 111, pp. 214-229. https://doi.org/10.1016/j.cor.2019.07.001

**Assignment problem with conflicts.** / Öncan, Temel (Corresponding author); Şuvak, Zeynep; Akyuz, Hakan; Kuban Altinel, I.

Research output: Contribution to journal › Article › Academic › peer-review

TY - JOUR

T1 - Assignment problem with conflicts

AU - Öncan, Temel

AU - Şuvak, Zeynep

AU - Akyuz, Hakan

AU - Kuban Altinel, I.

PY - 2019/11

Y1 - 2019/11

N2 - We focus on an extension of the assignment problem with additional conflict (pair) constraints in conjunction with the assignment constraints and binary restrictions. Given a bipartite graph with a cost associated with each edge and a conflict set of edge pairs, the assignment problem with conflict constraints corresponds to finding a minimum weight perfect matching without any conflicting edge pair. For example, some chemicals cannot be processed on close processors, food and toxic products cannot be stored neighboring locations at the same storage area, and machines cannot be sent to process jobs without satisfying some spatial constraints. Unlike the well-known assignment problem, this problem is NP-hard. We first introduce a realistic special class and demonstrate its polynomial solvability. Then, we propose a Branch-and-Bound algorithm and a Russian Doll Search algorithm using the assignment problem relaxations for lower bound computations, and introduce combinatorial branching rules based on the conflicting edges in an optimal solution of the relaxations. According to the extensive computational experiments we can say that the proposed algorithms are very efficient.

AB - We focus on an extension of the assignment problem with additional conflict (pair) constraints in conjunction with the assignment constraints and binary restrictions. Given a bipartite graph with a cost associated with each edge and a conflict set of edge pairs, the assignment problem with conflict constraints corresponds to finding a minimum weight perfect matching without any conflicting edge pair. For example, some chemicals cannot be processed on close processors, food and toxic products cannot be stored neighboring locations at the same storage area, and machines cannot be sent to process jobs without satisfying some spatial constraints. Unlike the well-known assignment problem, this problem is NP-hard. We first introduce a realistic special class and demonstrate its polynomial solvability. Then, we propose a Branch-and-Bound algorithm and a Russian Doll Search algorithm using the assignment problem relaxations for lower bound computations, and introduce combinatorial branching rules based on the conflicting edges in an optimal solution of the relaxations. According to the extensive computational experiments we can say that the proposed algorithms are very efficient.

KW - Assignment Problem

KW - Integer programming

KW - Branch and bound

KW - Conflicts

KW - Branch-and-bound

KW - Assignment problem

UR - http://www.scopus.com/inward/record.url?scp=85068505969&partnerID=8YFLogxK

U2 - 10.1016/j.cor.2019.07.001

DO - 10.1016/j.cor.2019.07.001

M3 - Article

VL - 111

SP - 214

EP - 229

JO - Computers & Operations Research

JF - Computers & Operations Research

SN - 0305-0548

ER -